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I'm studing probability, an I can see martingale everywhere. I can work with them, but I don't really see in what they are so important everywhere.

I know that $M_n$ is a martingale wrt the filtration $\mathcal F_n$ if $M_n$ is $\mathcal F_n-$measurable and if $$\mathbb E[M_{n+1}\mid \mathcal F_n]=M_n.$$ In what this definition makes them so important ? And why are they everywhere ?

user380364
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    Do you know how martingales originated? They came from gambling. They are models for what you call 'fair games'. However, they eventually became a powerful tool for proving theorems in Probability Theory. One of the nicest almost sure convergence theorems in Probability Theory is the Martingale Convergence Theorem. – Kavi Rama Murthy Aug 02 '18 at 08:40
  • See here: https://math.stackexchange.com/questions/908908/how-to-get-closed-form-solutions-to-stopped-martingale-problems – Math1000 Aug 02 '18 at 13:06
  • @KaviRamaMurthy This is a very good answer (+1). Their use in gambling and closed games is indisputable but why are they taught so extensively in Math Finance classes? Is it not the case that in most of the market "applications" martingale assumptions fail to be realized mostly due to unreachable incomplete information (ie. we do not know what we do not know). Isn't that a form of the ludic fallacy? (https://en.wikipedia.org/wiki/Ludic_fallacy#:~:text=The%20ludic%20fallacy%2C%20proposed%20by,world%20of%20games%20and%20dice%22.) – Pellenthor Oct 31 '20 at 16:00

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